Forcing and the Halpern-L\"auchli Theorem

TL;DR

This paper studies how various forcing notions affect the Halpern-L"auchli Theorem at inaccessible and measurable cardinals, showing preservation results under certain conditions.

Contribution

It establishes preservation of the Halpern-L"auchli Theorem under specific forcings for inaccessible and measurable cardinals, extending previous work on partition relations.

Findings

Preservation of the theorem by forcings of size less than  at inaccessible a.

Preservation of polarized partition relations under certain forcings.

Theorem preserved by < -closed forcings assuming is measurable.

Abstract

We investigate the effects of various forcings on several forms of the Halpern-L\"auchli Theorem. For inaccessible $\kappa$, we show they are preserved by forcings of size less than $\kappa$. Combining this with work of Zhang in \cite{Zhang17} yields that the polarized partition relations associated with finite products of the $\kappa$-rationals are preserved by all forcings of size less than $\kappa$ over models satisfying the Halpern-L\"auchli Theorem at $\kappa$. We also show that the Halpern-L\"auchli Theorem is preserved by ${{<}\kappa}$-closed forcings assuming $\kappa$ is measurable, following some observed reflection properties.